17 APPLICATIONS: II DUAL OF L1 203
(a) Let x0 e X. Show that, for every complex-valued continuous function g on X
the partial sums sn(g) are bounded (by a number depending on g and on x0) if and only
if the set of numbers Hn(*0) is bounded (use Problem I (a) and the Banach-Steinhaus
theorem). (This result generalizes Problem 2(a) of Section 11.6.)
(b) For every complex-valued continuous function g on X the partial sums sn(g)
converge uniformly to g if and only if the following two conditions are satisfied: (1) every
complex-valued continuous function on X can be approximated uniformly on X by
linear combinations of the/n; (2) there exists a constant a such that |Hrt(;t)| :g a for all
jc e X and all n. (To prove that (2) is necessary, observe that if sn(g) converges uniformly
to g, then for any increasing sequence of integers (nk) and any sequence (xk) of points
of X, the sequence of numbers Knk(xk, y)g(y) dp<(y) is bounded (by a number depend-
ing on g); now use the Banach-Steinhaus theorem.)
In particular, for the Haar orthonormal system (Section 8.7, Problem 7), the
Lebesgue functions are uniformly bounded.
3. Let X be a locally compact space, ft a positive measure on X, and p a real number
satisfying l<^/?<+°o; let q=p/(p—l). Let g be a complex-valued measurable
function such that fg is integrable for all /e ^(X, p). Show that ge -S?£,(X, ft).
(Show that the mapping f\-+?g of Lp into L1 is continuous, by using the closed graph
theorem (12.16.11).) (Cf. Section 12.16, Problem 240
4. Let U be a continuous endomorphism of L£(X, ft) which satisfies the conditions of
Section 13.11, Problem 16. Its transpose 1U is therefore a continuous endomorphism
of the Banach L£(X, ft) with norm :g 1 (Section 12.15, Problem 4) and such that the
relation /^ 0 implies '£/-/;>0. For any /e J2?£(X, ft), we denote by *U-f any
function in the class *(J'• /
(a) Let (/„) be a sequence of functions in &* which converges almost everywhere
to/and is such that the sequence of norms N «,(/.) is bounded. Show that the sequence
CU'fn) converges almost everywhere to '£/•/. (Consider the sequence of functions
gn — sup|/n+p — /„!, and use the relation \(?U• g^hdp— \gn(U-h)d^ which is
valid for any function h e J2?£.)
(b) Let W ^ 0 be a function in '&*. With the notation of Section 13.11, Problem 16,
show that if *U • *F <[ *F almost everywhere, then for each function /e J^i, we have
I /SPdft 2> 0. (Consider the measure *F • ft and apply Section 13.11, Problem 16(d).)
J E(/)
(c) With the notation of Section 13.11, Problem 18, show that *U • (pxQ^.<pxQ
almost everywhere. Deduce that, for each function /e JS?£ which vanishes almost
everwhere in X0, the function U -/vanishes almost everywhere in X0 .
5. With the notation of Section 13.11, Problem 18, suppose that the set X0 is negligible.
(This will always be the case if ft is bounded and U • 1 = 1.)
(a) Show that if/e ^ is such that '£/•/</ almost everywhere, then <£/•/=/
almost everywhere. (Majorize the integral of the function
(/- *u •/)($ + u • ® + • • • + un - oj.)
(b) Let / be the set of all functions h e £f% such that '£/ • h = h almost everywhere
(i.e., the set of functions whose classes h are fixed by tU). Show that the constant
functions belong to / and that if (hn) is any sequence in / such that the sequence
(Noo(/zn)) is bounded, then sup hn e /. (Use (a) above, and Problem 4(a).) Deduce> w(z) for all z e K. Show that S is the set of points (z, £) e E x R such thateach n a measure v e K such that